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From: Richard Damon <richard@damon-family.org>
Newsgroups: comp.theory
Subject: Re: How to write a self-referencial TM?
Date: Sat, 17 May 2025 16:07:14 -0400
Organization: i2pn2 (i2pn.org)
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On 5/17/25 3:55 PM, olcott wrote:
> On 5/17/2025 2:46 PM, Mike Terry wrote:
>> On 17/05/2025 20:26, wij wrote:
>>> On Sat, 2025-05-17 at 15:45 +0100, Mike Terry wrote:
>>>> On 17/05/2025 04:01, wij wrote:
>>>>> On Fri, 2025-05-16 at 23:51 +0100, Mike Terry wrote:
>>>>>> On 16/05/2025 20:35, wij wrote:
>>>>>>> On Fri, 2025-05-16 at 16:33 +0100, Mike Terry wrote:
>>>>>>>> On 16/05/2025 12:40, wij wrote:
>>>>>>>>> On Fri, 2025-05-16 at 03:26 +0100, Mike Terry wrote:
>>>>>>>>>> On 16/05/2025 02:47, wij wrote:
>>>>>>>>>>> On Fri, 2025-05-16 at 01:40 +0100, Mike Terry wrote:
>>>>>>>>>>>> On 15/05/2025 19:49, wij wrote:
>>>>>>>>>>>>> On Thu, 2025-05-15 at 17:08 +0100, Mike Terry wrote:
>>>>>>>>>>>>>> On 14/05/2025 18:53, wij wrote:
>>>>>>>>>>>>>>> On Wed, 2025-05-14 at 12:24 -0500, olcott wrote:
>>>>>>>>>>>>>>>> On 5/14/2025 11:43 AM, wij wrote:
>>>>>>>>>>>>>>>>> On Wed, 2025-05-14 at 09:51 -0500, olcott wrote:
>>>>>>>>>>>>>>>>>> On 5/14/2025 12:13 AM, wij wrote:
>>>>>>>>>>>>>>>>>>> Q: Write a turing machine that performs D function 
>>>>>>>>>>>>>>>>>>> (which calls
>>>>>>>>>>>>>>>>>>> itself):
>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>> void D() {
>>>>>>>>>>>>>>>>>>>           D();
>>>>>>>>>>>>>>>>>>> }
>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>> Easy?
>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>> That is not a TM.
>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>> It is a C program that exists. Therefore, there must be 
>>>>>>>>>>>>>>>>> a equivalent TM.
>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>>> To make a TM that references itself the closest
>>>>>>>>>>>>>>>>>> thing is a UTM that simulates its own TM source-code.
>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>> How does a UTM simulate its own TM source-code?
>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>> You run a UTM that has its own source-code on its tape.
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>> What is exactly the source-code on its tape?
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> Every UTM has some scheme which can be applied to a (TM & 
>>>>>>>>>>>>>> input tape) that is to
>>>>>>>>>>>>>> be
>>>>>>>>>>>>>> simulated.
>>>>>>>>>>>>>> The
>>>>>>>>>>>>>> scheme says how to turn the (TM + input tape) into a 
>>>>>>>>>>>>>> string of symbols that
>>>>>>>>>>>>>> represent
>>>>>>>>>>>>>> that
>>>>>>>>>>>>>> computation.
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> So to answer your question, the "source-code on its tape" 
>>>>>>>>>>>>>> is the result of
>>>>>>>>>>>>>> applying
>>>>>>>>>>>>>> the
>>>>>>>>>>>>>> UTM's
>>>>>>>>>>>>>> particular scheme to the combination (UTM, input tape) 
>>>>>>>>>>>>>> that is to be simulated.
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> If you're looking for the exact string symbols, obviously 
>>>>>>>>>>>>>> you would need to
>>>>>>>>>>>>>> specify
>>>>>>>>>>>>>> the
>>>>>>>>>>>>>> exact
>>>>>>>>>>>>>> UTM
>>>>>>>>>>>>>> being used, because every UTM will have a different answer 
>>>>>>>>>>>>>> to your question.
>>>>>>>>>>>>>>
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> Mike.
>>>>>>>>>>>>>
>>>>>>>>>>>>> People used to say UTM can simulate all TM. I was questing 
>>>>>>>>>>>>> such a UTM.
>>>>>>>>>>>>> Because you said "Every UTM ...", so what is the source of 
>>>>>>>>>>>>> such UTM?
>>>>>>>>>>>>
>>>>>>>>>>>> Yes, a UTM can simulate any TM including itself.  (Nothing 
>>>>>>>>>>>> magical changes when a
>>>>>>>>>>>> UTM
>>>>>>>>>>>> simulates
>>>>>>>>>>>> itself, as opposed to some other TM.)
>>>>>>>>>>>
>>>>>>>>>>> Supposed UTM exists, and denoted as U(X), X denotes the tape 
>>>>>>>>>>> contents of the
>>>>>>>>>>> encoding of a TM. And, U(X) should function the same like X.
>>>>>>>>>>> Given instance U(U(f)), it should function like f from the 
>>>>>>>>>>> above definition.
>>>>>>>>>>> But, U(U(f)) would fall into a 'self-reference' trap.
>>>>>>>>>>
>>>>>>>>>> There is no self-reference trap.
>>>>>>>>>>
>>>>>>>>>> In your notation:
>>>>>>>>>>
>>>>>>>>>> -  f represents some computation.
>>>>>>>>>> -  U(f) represents U being run with f on its tape.
>>>>>>>>>>         Note this is itself a computation, distinct from f of 
>>>>>>>>>> course
>>>>>>>>>>         but having the same behaviour.
>>>>>>>>>> -  U(U(f)) represents U simulating the previous computation.
>>>>>>>>>>
>>>>>>>>>> There is no reason U(f) cannot be simulated by U.  U will have 
>>>>>>>>>> no knowledge that it is
>>>>>>>>>> "simulating
>>>>>>>>>> itself", and will just simulate what it is given.
>>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>> Mike.
>>>>>>>>>
>>>>>>>>> Sorry for not being clear on the UTM issue (I wanted to mean 
>>>>>>>>> several things in one post).
>>>>>>>>> You are right there is no self-reference.
>>>>>>>>> I mean 'UTM' is not a complete, qualified TM because the 
>>>>>>>>> contents of the tape
>>>>>>>>> would not be defined. Saying "UTM can simulate any TM" is 
>>>>>>>>> misleading because
>>>>>>>>> no such TM (UTM as TM) exists.
>>>>>>>>
>>>>>>>> What do you mean "the contents of the tape would not be 
>>>>>>>> defined"?  A TM is /equipped/ with
>>>>>>>> an
>>>>>>>> infinite tape, but the /contents/ of that tape are not a part of 
>>>>>>>> that TM's definition.
>>>>>>>>
>>>>>>>> For example we could build a TM P that decides whether a number 
>>>>>>>> is prime.  Given a number n,
>>>>>>>> we
>>>>>>>> convert n into the input tape representation of n, and run P 
>>>>>>>> with that tape as input.
>>>>>>>>
>>>>>>>> It's essentially no different for UTMs.  Such a UTM certainly is 
>>>>>>>> a "complete TM", equipped
>>>>>>>> with
>>>>>>>> its
>>>>>>>> own input tape.  Of course we don't know what's on the input 
>>>>>>>> tape because nobody has said
>>>>>>>> yet
>>>>>>>> what
>>>>>>>> computation we are asking it to simulate!  [Similarly we don't 
>>>>>>>> know what's on P's input
>>>>>>>> tape,
>>>>>>>> until
>>>>>>>> we know what n we want it to test for primeness.]  Once you say 
>>>>>>>> what computation you want
>>>>>>>> the
>>>>>>>> UTM to
>>>>>>>> simulate we can build a tape string to perform that particular 
>>>>>>>> simulation.  That is the case
>>>>>>>> /whatever/ computation we come up with, so it is simply the case 
>>>>>>>> [not misleading] that the
>>>>>>>> UTM
>>>>>>>> can
>>>>>>>> simulate any computation.
>>>>>>>>
>>>>>>>>
>>>>>>>> Mike.
>>>>>>>
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